
where the final equation has also utilized the jump conditions of Eq. (REF ).
As noted in Ref. [1]}, integrating Eq. (), we end up with a tautology and end up with boundary terms that provide no additional physical information. By moving the Dirac deltas of Eq. () to the LHS and evaluating the derivatives, we derive an alternative form of the equation which lacks Dirac deltas (implicit or explicit):
\(\dfrac{e^{-2A}}{\sqrt{6}}\,\phi _{0}^{\prime }\,\hat{f}^{\,\prime } = e^{2A} \, r_{c}^{2} \, (\square \hat{r}) + 4 A^{\prime }\,\hat{r}^{\,\prime } + \dfrac{1}{6} (\phi _{0}^{\prime })^{2}\,\hat{r} + \sqrt{\dfrac{8}{3}}\,e^{-2A}\,\dot{V}r_{c}^{2}\,\hat{f}~. \) 
